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%\author{2024级应用统计学、2024级数学与应用数学 }
\author{王立庆（2023级数学与应用数学1班） }
\title{数学物理方法教学大纲 }
%\date{2025年3月22日}

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\section*{时间地点}
\begin{itemize}
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\item 上课时间地点：周二下午5-6节，六教210；周四上午1-2节，六教210.
\item 答疑时间地点：周三下午15:00 - 17:00, 周三晚上18:30 - 20:00, 一教210. 

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\section*{使用教材}
\begin{enumerate}\itemsep0em 
\item  K. F. Riley, M. P. Hobson, S. J. Bence. Mathematical Methods for Physics and Engineering. Cambridge University Press. Third Edition. 2006. 
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%\section*{主要内容}
\section*{学习章节}
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\begin{itemize}\itemsep0em 

\item  第24章：复变函数。（8周）

\item  第20章：偏微分方程。（4周）

\item  第21章：分离变量法、积分变换法。（3周）


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\section*{参考教材}
\begin{enumerate}\itemsep0em 
\item  梁昆淼. 数学物理方法. 高等教育出版社, 2020年11月第五版。
\item  上海交通大学数学系. 数学物理方法.  上海交通大学出版社, 2016年9月第2版。
\item  上海交通大学数学系. 数学物理方法试题分析与解答.  上海交通大学出版社, 2014年2月第1版。

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\section*{课程成绩}
\begin{enumerate}
\item  平时成绩 50 \%. %, 包括课堂考勤、课外作业、阶段测验、。
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\item[1.1.] 课堂考勤10次，共20分。
\item[1.2.] 课外作业11次，共30分。
\item[1.3.] 阶段测验3次，共30分。
\item[1.4.] 期中考试1次，共20分。
\end{enumerate}

\item  期末考试 50 \%. %, 选择题、填空题、计算题和证明题。
%\begin{enumerate}
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%\item[2.1.] 填空题10个，共20分。
%\item[2.2.] 计算题10个，共60分。
%\item[2.3.] 证明题2个，共20分。
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\section*{授课计划}
 
\begin{tabular}{|p{0.8cm}|p{0.8cm}|p{0.8cm}|p{5cm}|p{3cm}|p{2cm}|}  \hline 
课次	&周	&章节	&内容	&布置作业 &实践 \\ \hline \hline
1	&1	&24.1	&单变量复变函数	&作业1  & \\ \hline 
2	&1	&24.2	&柯西-黎曼方程		&  & \\ \hline 
3	&2	&24.3	&复变量的幂级数	&作业2  & \\ \hline 
4	&2	&24.4	&一些初等函数		&  &编程实践 \\ \hline 
5	&3	&24.5	&多值函数与分枝	&作业3  &\\ \hline 
6	&3	&24.6	&复变函数的奇点与零点	&  & \\ \hline 
7	&4	&	&测验 & & \\ \hline 
8	&4	&	&复习 & & \\ \hline \hline 
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9	&5	&24.7	&共形映照			&作业4 &编程实践 \\ \hline 
10	&5	&24.8	&复积分			& & \\ \hline 
11	&6	&24.9	&柯西定理			&作业5 & \\ \hline 
12	&6	&24.10	&柯西积分公式		& & \\ \hline 
13	&7	&24.11	&泰勒级数与洛朗级数	&作业6 & \\ \hline 
14	&7	&24.12	&留数定理			& & \\ \hline 
15	&8	&24.13	&计算定积分		& & \\ \hline 
16	&8	&	&期中考试 & & \\ \hline \hline 
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17	&9	&20.1	&一些重要的偏微分方程	&作业7 & \\ \hline 
18	&9	&20.2	&解的一般形式		& & \\ \hline 
19	&10	&20.3	&通解与特解		&作业8 & \\ \hline 
20	&10	&20.4	&波动方程			& &编程实践 \\ \hline 
21	&11	&20.5	&扩散方程			&作业9 & \\ \hline 
22	&11	&20.6	&特征与解的存在性		& & \\ \hline 
23	&12	&20.7	&解的唯一性			& & \\ \hline 
24	&12	&	&测验 & & \\ \hline \hline 
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25	&13	&21.1	&分离变量法		&作业10  & \\ \hline 
26	&13	&21.2	&分离解的叠加		& &编程实践 \\ \hline 
27	&14	&21.3	&极坐标形式的分离变量法	&作业11 & \\ \hline 
28	&14	&21.4	&积分变换法		& & \\ \hline 
29	&15	&21.5	&非齐次方程、格林函数	& & \\ \hline 
30	&15	&	&测验 & &  \\ \hline \hline 
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	&16	&	&期末考试 & & \\ \hline 
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\end{tabular}


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\section*{教学目标}

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\subsection*{第24章：复变函数}
\begin{tabular}{|p{0.8cm}|p{4cm}|p{10cm}|}  \hline 
章节 &标题 & 重点和难点 \\ \hline 
24.1 &单变量复变函数&理解解析函数的概念。验证给定函数是否为解析函数。	\\ \hline 
24.2 &柯西-黎曼方程&导出柯西-黎曼方程。使用柯西-黎曼方程证明其它结论。	\\ \hline 
24.3 &复变量的幂级数&计算复变量的幂级数的收敛半径。	\\ \hline 
24.4 &一些初等函数&掌握指数函数、对数函数和幂函数的定义和性质。	
\newline 编程实践：编程作图画出初等函数的映照图像。	\\ \hline 
24.5 &多值函数与分枝&找出给定复变函数的分枝点和单值区域。	\\ \hline 
24.6 &复变函数的奇点与零点&区分可去奇点、极点和本性奇点。计算极点和零点的阶数。	\\ \hline 
24.7 &共形映照&描述分式线性变换对应的共形映照。
\newline 编程实践：编程作图把多角形共形变换到圆形。 \\ \hline 
24.8 &复积分&理解复积分的概念。计算一些复积分的例子。	\\ \hline 
24.9 &柯西定理	&证明和使用柯西定理。	\\ \hline 
24.10 &柯西积分公式&证明和使用柯西积分公式。	\\ \hline 
24.11 &泰勒级数与洛朗级数&将给定复变函数展开成泰勒级数或洛朗级数。	\\ \hline 
24.12 &留数定理&证明和使用留数定理。	\\ \hline 
24.13 &计算定积分&使用复积分的方法计算实函数的定积分。	
\\ \hline 
\end{tabular}

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\subsection*{第20章：偏微分方程}
\begin{tabular}{|p{0.8cm}|p{4cm}|p{10cm}|}  \hline 
章节 &标题 & 重点和难点 \\ \hline 
20.1 &一些重要的偏微分方程&导出一维波动方程和扩散方程。
\newline 了解拉普拉斯方程、泊松方程、薛定谔方程。	\\ \hline 
20.2 &解的一般形式&验证一个偏微分方程的解可以包含一个任意函数。	\\ \hline 
20.3 &通解与特解&求一些一阶和二阶偏微分方程的通解。
\newline  求满足给定边值条件的特解。	\\ \hline 
20.4 &波动方程&波动方程的解的物理含义。求解三维波动方程。
\newline 编程实践：编程作出一维和二维波动方程的解的动画。	\\ \hline 
20.5 &扩散方程	&根据物理含义通过变量代换将一维扩散方程化为常微分方程。
\newline 计算红外激光在绝缘板上的能量扩散的例子。  \\ \hline 
20.6 &特征与解的存在性&计算一阶和二阶偏微分方程的特征曲线。
\newline 理解不同类型的边值条件。 	\\ \hline 
20.7 &解的唯一性&证明泊松方程在两类边值条件下的解的唯一性。 \\ \hline 
\end{tabular}

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\subsection*{第21章：分离变量法与积分变换法 }
\begin{tabular}{|p{0.8cm}|p{4cm}|p{10cm}|}  \hline 
章节 &标题 & 重点和难点 \\ \hline 
21.1 &分离变量法&使用分离变量法求解一维扩散方程和二维拉普拉斯方程。	\\ \hline 
21.2 &分离解的叠加&计算矩形金属板的稳态温度分布。 
\newline 计算一端稳定加热的长杆的稳态温度分布。
\newline 编程实践：编程作出一维和二维扩散方程的解的动画。	\\ \hline 
21.3 &极坐标形式的分离变量法&计算圆柱体内的稳态温度分布。	\\ \hline 
21.4 &积分变换法&使用拉普拉斯变换计算管道内的盐溶液的扩散总量。	
\newline  使用傅立叶变换计算金属杆的温度分布。 \\ \hline 
21.5 &非齐次方程、格林函数&使用格林函数求解非齐次偏微分方程。	 \\ \hline 
\end{tabular}


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